The edge of a cube is increasing at a rate of 7 cm/s. The rate of change of area of the cube when edge of the cube is 3 cm is:

$252\, cm^2/s$

The correct answer is Option (4) → $252\, cm^2/s$

Let the edge of the cube be $x$ cm.

Surface area of cube: $A = 6x^2$

Given: $\frac{dx}{dt} = 7 \, \text{cm/s}$

Differentiate w.r.t. time $t$:

$\frac{dA}{dt} = 12x \frac{dx}{dt}$

Substitute $x = 3$ and $\frac{dx}{dt} = 7$:

$\frac{dA}{dt} = 12(3)(7) = 252$

Final Answer:

$\frac{dA}{dt} = 252 \, \text{cm}^2/\text{s}$